Statistics shows Expected Value, Variance, Std Deviation and Mode for the current parameters; Show Formulas prints the rule under each card. Learn more about the statistics panel
PDF, the first tab above the chart, draws the density curve f(x); the red line marks E[X]. Learn more about the PDF curve
In the Probability Calculator, tap P(X < x), P(X ≤ x), P(X > x) or P(X ≥ x) and type x; strict and non-strict tabs give the same number. Learn more about the probability calculators
The calculator opens on P(a ≤ X ≤ b): type a and b for the area between them; the other three range tabs give the same area. Learn more about the empirical rule
Use the μ (mu) slider to shift the distribution's center along the x-axis. Values range from -10 to 10, moving the bell curve left or right without changing its shape.
Adjust σ (sigma) slider to control spread. Smaller σ creates a tall, narrow curve with values concentrated near μ. Larger σ produces a short, wide curve with more variability.
Watch how changing these parameters affects the visualization. The curve always remains symmetric around μ, with the same bell shape regardless of parameters - only location and scale change.
Reading the Statistics Panel
The Statistics panel sits in the left column, under Parameters. Its four cards show Expected Value, Variance and Std Deviation to four decimals, and Mode to four decimals. Every card recomputes on each slider move, so the panel always describes the curve in the middle column; the red E[X] line on the chart sits at the Expected Value number.
Show Formulas prints the rule under each card: E[X]=μ, Var(X)=σ2, σ and Mode=μ. The button then reads Hide Formulas. The numbers are the same either way; the formulas only show where they come from.
At the opening values, μ=0, σ=1, the cards read 0.0000, 1.0000, 1.0000 and 0.0000. Expected Value and Mode both equal μ, and Std Deviation is σ itself. At μ=3, σ=2 the cards read 3.0000, 4.0000, 2.0000 and 3.0000.
The Parameters cards above the panel repeat each value with its symbol and a one-line meaning: Mean (μ), Standard Deviation (σ) and Variance (σ²). For where the formulas come from, see Distribution Properties and Statistics.
Reading the PDF Curve
DemoPDF, CDF and Table tabs
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The PDF (Probability Density Function) displays the characteristic bell-shaped curve. The peak occurs at x = μ, representing the distribution's mean, median, and mode - all equal due to symmetry.
The curve extends from -∞ to +∞ theoretically, but the calculator shows the practical range μ ± 4σ where 99.99% of probability mass resides. Areas under the curve represent probabilities.
Inflection points (where curvature changes) occur at μ ± σ. Between these points, the curve is steepest, containing about 68% of the total probability. This geometric feature helps visualize the 68-95-99.7 rule.
Understanding the CDF Display
The CDF (Cumulative Distribution Function) shows the characteristic S-curve rising from 0 to 1. At any point x, the height gives P(X ≤ x), the probability of values at or below x.
The CDF's steepest region occurs around μ where the PDF peaks. At μ itself, CDF equals 0.5 - exactly half the probability lies below the mean due to symmetry.
The CDF approaches but never quite reaches 0 or 1 at the display's edges. At μ ± 3σ, CDF values are approximately 0.0013 and 0.9987, leaving only 0.27% probability in the extreme tails combined.
Using the Table Tab
Table is the third tab above the chart, after PDF and CDF. It replaces the chart with three columns, x, f(x) and F(x), one row per point the chart plots, every value to six decimals. The rows follow the sliders: move one and the whole table recomputes.
Use it where the chart is too small to read. At the opening values, μ=0, σ=1, the table runs from x=−4 to x=4, that is μ±4σ, in steps of about 0.0267. The end rows have f(x)=0.000134, and F(4)=0.999968. Set σ=2 and the rows span −8 to 8.
The F(x) column is the CDF as numbers, and f(x) is the height of the PDF curve at that x. The density is not a probability; for one, use an interval. For a value the table does not list, or a range of values, use the Probability Calculator below the chart.
The Explorer at Its Opening Parameters
The standard normal, μ=0 and σ=1, is drawn over the window μ±4σ. The density peaks at f(0)=2π1=0.3989, with the red mean line through the centre of the bell.
The CDF tab shows the sigmoid crossing 0.5 exactly at the mean, and reaching 0.99997 at the right edge.
Normal, mu = 0, sigma = 1: PDF above, CDF below
The bell peaks at 1/sqrt(2 pi) = 0.3989 and the sigmoid below crosses 0.5 at the mean. Shrink sigma and the peak rises without limit: a density may exceed 1, a probability may not.
The peak height is the clearest illustration that a density is not a probability. Here it is 0.399; shrink σ and it rises without limit, because the area underneath must stay at 1. A density above 1 is perfectly legal, a probability above 1 is not.
The two tabs locate σ differently, which is why both are worth looking at. On the PDF it is the distance from the peak to the inflection points; on the CDF it is where the curve stops steepening. Same number, two different visual cues.
Because the Gaussian has no elementary antiderivative, the CDF cannot be written in closed form with ordinary functions — the tool evaluates it through an error-function approximation, which is why every normal probability you have ever looked up came from a table or a numerical routine rather than a formula.
Using the Probability Calculators
DemoCumulative probabilities
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Enter value x in the P(X ≤ x) calculator to find the area under the curve to the left of x. This uses the error function erf(z) where z = (x - μ)/σ is the standardized Z-score.
P(X ≥ x) gives the right-tail probability, computed as 1 - P(X ≤ x). This answers questions like "What fraction of values exceed x?"
For range probabilities P(a ≤ X ≤ b), take CDF(b) - CDF(a) from two P(X ≤ x) readings, giving the area under the curve between a and b. This is the fundamental probability for any continuous interval.
Working with Z-Scores
Every normal distribution calculation uses standardization: Z = (X - μ)/σ transforms values to standard normal (μ = 0, σ = 1). The calculator performs this conversion automatically.
Z-scores measure distance from the mean in standard deviation units. Z = 2 means "2 standard deviations above the mean." Z = -1.5 means "1.5 standard deviations below."
The standard normal table (Z-table) provides CDF values for Z-scores. For any normal distribution, standardize first, then look up the Z-score. This is why one table suffices for all normal distributions.
The Empirical Rule (68-95-99.7)
DemoThe empirical rule
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The 68% rule: About 68% of values fall within μ ± σ. For μ = 100, σ = 15, expect 68% of values in [85, 115].
The 95% rule: About 95% fall within μ ± 2σ. Using the same parameters, 95% lie in [70, 130]. Only 5% of values fall more than 2 standard deviations from the mean.
The 99.7% rule: About 99.7% fall within μ ± 3σ, or [55, 145]. Values beyond ±3σ are rare, occurring less than 0.3% of the time. This rule underpins control charts and outlier detection.
Key Properties and Real-World Applications
The right column holds two lists. Key Properties has 6 rows, each ending in ▼: Support: X ∈ (-∞, ∞), Symmetric Bell Curve, 68-95-99.7 Rule (Empirical Rule), Central Limit Theorem, Standardization to Z-Score, Maximum Entropy Distribution. Row titles here are those at the opening values. Tap a row to open its explanation; tap another and the first closes, so one explanation is open at a time; tap the open row to close it.
Symmetric Bell Curve and 68-95-99.7 Rule (Empirical Rule) quote the current μ and σ; the second lists the intervals μ±σ, μ±2σ and μ±3σ as numbers. Open a row, move a slider, and the numbers in the open explanation follow.
Real-World Applications, below, lists 6 situations the normal distribution models, starting with Heights and weights of populations and Measurement errors in scientific experiments. This list is fixed text and does not change with the sliders.
The normal (Gaussian) distribution is the most important continuous probability distribution, modeling natural phenomena that result from many small, independent factors. Its bell shape appears throughout nature, science, and statistics.
Two parameters completely define it: μ (mean) determines location, σ (standard deviation) determines spread. The distribution is symmetric around μ, with probability decreasing exponentially as distance from μ increases.
The normal distribution is fundamental because of the Central Limit Theorem: sums and averages of independent random variables tend toward normality regardless of the original distribution. For comprehensive theory, see normal distribution theory page.
Central Limit Theorem Connection
The Central Limit Theorem explains why normal distributions appear so frequently. When you sum or average many independent random variables, the result approaches a normal distribution even if the original variables aren't normal.
This applies to measurement errors (sum of many small perturbations), test scores (sum of knowledge across many topics), and manufacturing tolerances (sum of variations in multiple processes).
The theorem works surprisingly quickly - often just 30 observations suffice for reasonable normality. This makes the normal distribution applicable even when underlying processes aren't purely Gaussian.
Distribution Properties and Statistics
Mean, median, and mode all equal μ due to perfect symmetry. There's no skewness - the distribution is perfectly balanced around its center.
Variance equals σ², the square of the standard deviation. Standard deviation σ is the natural scale parameter, representing typical deviation from the mean.
The distribution has zero skewness (symmetric) and kurtosis of 3 (defining "normal" tail behavior). Distributions with kurtosis > 3 have heavier tails; kurtosis < 3 have lighter tails than normal.
Related Distributions and Calculators
The standard normal distribution (Z-distribution) has μ = 0 and σ = 1, serving as the reference for all normal calculations. Any normal distribution can be standardized using Z = (X - μ)/σ.
The t-distribution approximates normal for large sample sizes but has heavier tails, accounting for additional uncertainty when estimating σ from data. Use t for small samples (n < 30).
Related Tools:
Z-Score Calculator - Standardization and percentiles
T-Distribution Calculator - Small sample inference
Chi-Square Distribution - Related to normal's squared values
Hypothesis Testing Tools - Applications of normal distribution